Numerical Evaluation of Special Functions

D. W. Lozier and F. W. J. Olver

7. Future Trends

Great progress has been made in recent years in the
construction of software for generating the special functions,
yet enormous gaps remain for functions having variable parameters
in addition to the argument. This is especially true when the
variables are complex. In this concluding section we offer some
general suggestions concerning future work in this area.

First, because of the sheer magnitude of the effort
required, there should be a perceived physical or other applied
need before a decision is made to embark on the construction of
extensive new software for functions of two or more variables. At
present there are simply too many gaps to fill to be able to
indulge in the luxury of arbitrary selection. Moreover, great
care should be exercised in the choice of actual functions to be
generated. For example, neither the Airy function nor
the Bessel function of the second kind has a useful
role when the argument z is not real; compare
[ Olv74, Chapters 7 and 11]
.

Second, coverage of a chosen region should be dictated by
uniform accuracy requirements (in an appropriate measure), not by
the limitations of the methods that happen to be used. At the
very least it is frustrating for users to discover that the
precision yielded by a package varies widely, or worse still
disappears altogether, in parts of the claimed regions of coverage.

Third, the potential offered by the ongoing increase in
power of computers should be exploited with a view to reducing
the number and complexity of algorithms to be used. This
includes, for example, the use of parallel or vector methods for
summing series [ Kar91]
or solving differential or difference
equations [ LO93]
.

Fourth---and here we are looking further into the future---the
use of systems of computer arithmetic other than floating-point
should be considered. The floating-point system has two
disadvantages which become especially annoying and time-consuming in the
construction of special-function software. One is that the
associated error measure, relative precision, is quite
inappropriate in the neighborhoods of zeros. The other stems
from failure due to overflow or underflow: here the usual remedy
of rescaling can be difficult to apply, owing to the extremely
varied asymptotic behavior of functions of several variables. A
system of computer arithmetic that is capable of overcoming both
problems in an elegant manner is the so-called level-index system
[ COT89]
.

Lastly, any new algorithm or package should be documented
fully. It should also be subjected to exhaustive testing
procedures, and these, too, need to be documented. Indeed, the
proposed testing procedures should be considered at an early
stage in the planning of the main algorithms. There are so many pitfalls in the
construction of algorithms for the special functions that the use
of undocumented or insufficiently tested packages is a risky
proposition.

Abstract:

This document is an excerpt from the current hypertext version of
an article that appeared in Walter Gautschi (ed.),
Mathematics of Computation 1943--1993: A Half-Century of
Computational Mathematics, Proceedings of Symposia in
Applied Mathematics 48, American Mathematical Society,
Providence, RI 02940, 1994.
The symposium was held at the University of British Columbia
August 9--13, 1993, in honor of the fiftieth anniversary of
the journal Mathematics of Computation.

The original abstract follows.

Higher transcendental functions continue to play varied and
important roles in investigations by engineers, mathematicians,
scientists and statisticians.
The purpose of this paper is to assist in locating useful approximations
and software for the numerical generation of these functions, and to
offer some suggestions for future developments in this field.